The cosmological constant: Einstein's fatal mistake that led to the Nobel Prize in Physics

Part I: From Einstein's Static Universe to De Sitter's Expanding and Accelerating Empty Universe

Albert Einstein said in a letter to his good friend from Holland, Paul Ehrenfest, on February 4, 1917, that "I have once again done something in the theory of gravitation that threatens to keep me in quarantine in a hospital for the insane. I hope you don't have one in Lidan, so I can visit you safely."

Einstein really had reason to worry about the reaction of his colleagues. He stepped up to the podium at the Prussian Academy of Sciences, and announced to his colleagues who looked at him in astonishment: "The fact is, I have come to the conclusion that the field equations of gravity that I have presented so far need a slight correction, so that the general theory of relativity avoids the fundamental difficulties that we... have shown to be applied to Newton's theory." The correction was the cosmological constant.

Let us first look at Newton's universe. We will think of it as an infinite universe. What will happen then?
The mass density of the universe will tend to zero. Einstein thought that over time the random movement of the stars moving in this universe would create the billiard ball effect when the matter would disappear into infinity: one star after another would jump in a jump "big enough to send this star on its course to infinity, while it could never return". If we wait long enough this Newtonian universe will eventually be completely empty: "The disappearance of density at infinity... hints at the disappearance of density at the center".

But our universe exists and as we can see for ourselves, it is dense, rich and packed with matter - the sun, our planets and millions of stars woven into the fabric of the Milky Way. Newton's version - or rather to say - the description of the universe's behavior on the scale of the universe as developed by Newton's successors in accordance with the mechanistic worldview using his physics, clearly cannot describe what is actually seen somewhere in the universe.

Thus Einstein sought to create his own replacement model to replace Newton's. Can these contradictions be reconciled when proposing an alternative model of the universe according to the field equations of Einstein's original theory of general relativity which he presented in 1915? Are the contradictions settled within general relativity when proposing a model of an infinite universe? Einstein understood that the answer was no; And that is why he proposed to the Prussian Academy an alternative he called the field equations of general relativity from 1915 + the cosmological constant and a static universe.

The alternative that Einstein proposed and the correction to the equations of relativity that followed it is the following:
Einstein initially proposed his version of the structure of the universe. He made two critical assumptions:
1. The universe contains an average density of matter that is "everywhere the same and different from zero".
2. It is static, without change in its structure over time. That is, when averaging over many stars, the universe is uniform and essentially static.

Expressing these two assumptions mathematically, Einstein derived a solution to his field equations from general relativity that would describe the behavior of this model of the entire universe. The solution described a state of unstable equilibrium. In simple language, the universe that emerges from this solution is limitless, but it is finite. It exists as a type of four-dimensional sphere, a "hypersphere". The surface of the sphere has no boundaries, no end, but it has a finite area or a finite volume. You can move around and around on the surface of the ball and never fall. There is so much ground to cover and more.

Einstein's universe has no end and encompasses a finite volume of space-time. Within this volume lies all the entire matter and all the energy of the universe, so that the universe also has a finite and positive mass density (or finite mass-energy), and this corresponds to the first of Einstein's conditions. The total sum of matter and energy created a gravitational field in the universe, which determined the shape of space-time for the entire system: a universe formed on a curved four-dimensional surface. Note that in such a universe a ray of light has to return to itself. In other words, a star should be visible not only in one direction but also in the exact opposite direction, if the dimensions of the universe were not so large and the starlight was not so dim.

As it has progressed so far Einstein's concept seems fine. But one problem remains... when Einstein's universe is left alone, it cannot be left alone. When it is assumed that there are no external forces acting, the total gravity of a system will cause it to collapse in on itself. Or alternatively, some initial outward compression or evolutionary change in the curvature of spacetime could have driven the expansion of the universe across time. One thing was clear, without correcting the field equations of general relativity from 1915, Einstein's universe could not remain fixed in place as it is, unchanged, over time.

The idea of ​​a dynamic universe expanding in time, one with a constant flux, was something of an abomination to Einstein. First, the Torah should look at the world as it is presented to it by observers. The observed low relative velocities of the stars led Einstein to the assumption, which was almost inevitable in 1917, that the universe is static.

Second, in 1917 very few physicists realized how big the universe was. Our galaxy - the Milky Way - was still generally seen as much as there was, and was seen as mostly static. In fact the Milky Way was understood as the entire universe. It is true that Shosto Melvin Sleeper, while observing the spectra from the spiral nebulae in 1910, discovered a preference for redshifts, of the types that would be created by propagation by the Doppler effect, but no one at that time knew what the spiral nebulae were. We had to wait another 13 or 14 years until Edwin Hubble studied the Andromeda Nebula and then they realized that the spiral nebulae are actually distant galaxies, star clusters that are even outside our galaxy. It is not known at all whether Einstein was aware of Slipper's redshifts in 1917. But it is clear that Einstein knew about something else that could produce a redshift of spectral lines: the gravitational field.

Einstein therefore corrected his universe problem by adding a new term to his gravity equations of November 1915, the "cosmological constant". As Einstein pointed out in his 1917 paper, the cosmological constant is required because of the assumption that the universe that Einstein proposed is static, not because it is finite. The cosmological constant served as an expression of a kind of anti-gravity, a force that pushes outwards between the masses in the gravitational fields that pull them inward. It acted as a repulsive force that increased with distance while the attractive force of gravity decreased with distance. Although there is a critical mass density where this repulsive force just balances the gravitational pull, the balance is unstable. A slight expansion will increase the repulsive force and decrease the attractive force so that the expansion will accelerate. This is a difficulty that Einstein should have seen when he proposed the cosmological constant! In the language of general relativity, the cosmological constant reverses the direction of the curvature of space-time created by the gravitational field; However, even in this formulation the same problem remains.

The correction was small and it does not seem to change the conclusions of general relativity about the orbit of Mercury and the like. But it had no basis in any practical physical idea or in any real-world observation. The constant was, as Einstein understood, "currently unknown... and is not justified by our practical knowledge of gravity". But still he got the job done. While adding the cosmological constant to general relativity, Einstein's universe was saved in a good way, it stood still, and did not change throughout eternity (almost...).

Needless to say, the chance of verifying the effect that Einstein described by astronomical means - the "double presence" of distant stars - and thus verifying Einstein's model of the universe - was extremely small. Using the best telescopes at the time, astronomers could see about ten thousand light years, while Einstein's conclusions, which were based on accepted data on the distribution of stars, assumed that the universe had a radius a thousand times greater - that is, ten million light years.

That exact year in 1917, Wilhelm de Sitter proposed an alternative solution to the Einstein equations + the cosmological constant. De Sitter's cosmological model was not static - the expansion of the de Sitter universe was accelerating. As I recall, Einstein held very stubbornly to two beliefs regarding his universe that guided him in building his cosmological model: the universe is static and its metric structure is completely determined by matter, meaning that its metric field provides what he called a year later in 1918 Mach's principle. And here de Sitter offers a vacuum solution to Einstein's field equations and thus offers a counterexample to this principle. There is no matter that would create the space-time curvature that the universe described. Einstein tried to get rid of the de Sitter position and attacked them by claiming that the solution is not static and that it is not free of matter after all. That is, Einstein was looking for matter underground in an acceleratingly expanding de Sitter universe. Einstein tried to find an error in the de Sitter model and he tried to claim that the de Sitter model has a singularity.

In March 1917 de Sitter sent Einstein a side-by-side comparison of the two models: his and Einstein's. He converted his "hyperboloid" model to new coordinates that are independent of time. Then the static form of the de Sitter solution was obtained. This shape in the spatial geometry of de Sitter's solution is the same as that of Einstein's solution - the "hypersphere" in four-dimensional Euclidean space. However, unlike Einstein's model, the temporal component of de Sitter's static matrix changes and disappears at the equator. And Einstein claimed that such a singular behavior of the metric is unacceptable and therefore it indicates the presence of matter at the equator.

Einstein was convinced that the singularity in the metric in the static coordinates is not only a byproduct of the mapping - writing the model in new coordinates, in coordinates that are used - but it seemed to Einstein that the threat posed by de Sitter's solution to Mach's principle was removed. De Sitter explained that the singularity is a by-product - use of the static coordinates and therefore not a true singularity. De Sitter's model did threaten Mach's principle.

The debate over de Sitter's and Einstein's models continued in Einstein's correspondence with the mathematicians Hermann Weill and Felix Klein. Finally, Felix Klein sent Einstein a letter in which he states, among other things, that the singularity at the equator in the static form of the de Sitter solution is a by-product of the way the time coordinate is introduced. The main theme of the letter was to tell Einstein that by transformation the singularity at the equator could be made to disappear and therefore it did not show at all the presence of matter in de Sitter's model.

In the end, after a few persuasions, Einstein accepted that the de Sitter solution is without matter, but he probably did not accept the de Sitter solution as a possible cosmological model. He still believed that any accepted cosmological model should be static. Einstein therefore had to accept that the de Sitter solution is a counterexample to Mach's principle, as formulated in 1918, and his criticism of the de Sitter solution requires correction. His field equations including the cosmological constant enabled matter-free solutions and a solution of an accelerating expanding universe. What do we do?... Wait for the next chapter.